Investigation of the optical spectra and spin-Hamiltonian parameters for the rhombic Cu²⁺ center in β-Ca₃(PO₄)₂

The four optical band positions and six spin-Hamiltonian parameters (the g factors g_i and the hyperfine structure constants A_i, i=x, y, z) for the rhombic Cu²⁺ center in β-Ca₃(PO₄)₂ are calculated by two theoretical methods: the complete diagonalization (of energy matrix) method and the perturbation theory method (PTM). Both methods are based on the cluster approach, in which the covalence due to admixture of ligands to the central metal ion is considered. The calculated results of the two methods are close to each other and agree with the experimental values. This suggests that both methods are effective in explaining the optical and electron paramagnetic resonance data for d⁹ ions in crystals. The calculations also show that although the admixture of |d_z²⟩ to the ground state wave function is small, it should not be neglected in calculations of spin-Hamiltonian parameters. So, in PTM, the high-order perturbation formulae of spin-Hamiltonian parameters based on the cluster approach for d⁹ ion in rhombic symmetry should be derived by taking into account the admixture of the |d_z²⟩ state to the |d_x²−y²⟩ ground state.